Specializing early

Question

Topic: this is a mathematics education question (but applies to other sciences too).

Assumptions: my first assumption is that most mathematical concepts used in research are not intrinsically more complicated to grasp than high-school and undergraduate maths, the main difference is the amount of prerequisites (and hence time and experience) involved. My second assumption is that some undergraduate topics currently taught compulsarily are a bit of a burden for someone focussed on a particular topic.

Now of course cognitive development is a constraint, but upon reaching the age of high-school, I would think that a fairly large proportion of the scientifically-enclined students could really understand things usually taught much later and indeed become active at research level within a few years, provided some shortcuts are introduced.

Early specialization: I'm wondering if a balanced curriculum already exists (or is planned) to provide such early specialization. What I'm looking for is this: a one-week panorama of maths (or physics, or biology) would be organized at the beginning, and then the students would decide which subtopic to study. For example someone interested by group theory (or quantum optics, or genetics) would thus start with basics at the age 15 or 16, and gradually learn more stuff and skills, but for a few years with a strong emphasis on things directly relevant for the chosen subtopic.

So for example the student specializing in group theory would only learn differential calculus and manifolds in passing in the context of Lie groups, and would skip most undergraduate real and functional analysis until it becomes relevant for his/her research topic, if at all. Of course other general courses would still be taught (history, sciences, programming, foreign languages...), but at least 50% of the student's week would be devoted to the research topic, ensuring satisfying progress.

Question: do you know of any active or planned educative curriculum (at a high-school or university, or maybe a specific home-schooling program) as outlined above? As an example of successful early specialization see e.g. the winners of the Siemens Foundation Prizes, but I haven't been able to learn much about their specific curriculum if any.

Note: Skipping grades in school to enter university earlier is not the point, I'm really interested in a subtopic-oriented curriculum.

Answer 1

As far as getting high school students involved in research by learning rapidly a narrow range of mathematics but in some depth, this is actually done in the mathematics section of the Research Science Institute program at MIT for students who have completed their junior year. Last year there were four projects in representation theory; I recall that one of them did not now linear algebra until some two weeks before the program (but learned quickly and completed a very successful project).

Sadly, I do not know of many other opportunities- RSI is a small program, and only a portion of it is for mathematics. I believe the PROMYS program supervises some research projects, but it is primarily for learning mathematics. Incidentally, many of the winners of competitions such as Siemens begin their projects at RSI.

Also, alumni of the RSI program do not necessarily end up specializing in the same fields that they did their projects (if they do eventually choose to pursue a career in mathematics, which does not always happen). It does give an exposure to a certain field, though.

Answer 2

I'm not sure how good an idea this would be. I happen to be in a position where I read many applications of students wishing to do a PhD in my group. Some applicants have a very definite idea of what it is that they want to do, but this is usually for lack of exposure to other topics. It is not unheard of that they end up doing their PhD is a completely different area.

I can speak from my own personal experience. At every stage in my life I was sure I knew what I wanted to study, but as I learnt and become exposed to new topics this changed; though not my certainty about my choice. Until about age 14 I wanted to study Molecular Biology. Had I specialised then I would not have seen any of the Physics and Mathematics which have become such an important part of my life.

The point I'm trying to make that is that early specialisation might be depriving the student from finding what it is they truly like. Freedom of choice is only ever meaningful if one can understand (or at least be aware of) the alternatives.

Of course, you could argue that there is a lot more information readily available to school children than when I went to school, so perhaps a more conscious choice can be made at an early age. There is however still a danger even within one discipline, say, Mathematics.

The research council which funds mathematical research in the UK commissioned an international review some years ago. A panel of respected non-UK mathematicians analysed the state of UK mathematical research. One of their conclusions was that due to the short length of the UK PhD (36 months at the time) students were forced to specialise earlier than in other countries (though not as early as the OP suggests) in order to complete their PhD in time. This then made it harder to switch fields later in their career and made them less competitive in the long run.

I will refrain from commenting here on the half-measures that were introduced to try to solve this problem, but I simply want to point out that even such a late "early specialisation" as this one is not desirable.

Answer 3

I agree with José's comment above: I do not think early specialisation is a good idea. Did I understand correctly that you want to give a one week to a 15-year old to decide on which area of mathematics to specialize?

I want to add something different, however. I fail to see how "some undergraduate topics currently taught compulsorily are a bit of a burden". Mathematics is not a set of disconnected areas. They are all highly related. Most research problems, while staying in one area, may be related to another, motivated by another, applicable in another, or steal ideas or techniques from another. One general course in, say, real analysis, complex analysis, abstract algebra, differential geometry, discrete mathematics, or topology is not a burden, but I dare say an actual necessity for anybody wanting to do research on any topic in pure math. To use your own example, someone doing research in Lie theory will benefit from, rather than be burdened by, a solid understanding of basic differential geometry. Or to use my own case, I am a Poisson geometer, but I have used ideas or results from all the above topics in my research.